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ArcSinh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Transformations > Related transformations > Differences involving the direct function > Involving tan-1(z)





http://functions.wolfram.com/01.25.16.0185.01









  


  










Input Form





ArcSinh[x] - ArcTan[y] == -2 I Pi (Floor[(-Arg[x + Sqrt[x^2 + 1]] - Arg[(1 - I y)^(-(I/2))] + Pi)/ (2 Pi)] + Floor[(Pi - Im[Log[x + Sqrt[x^2 + 1]]])/(2 Pi)] + Floor[((1/2) Re[Log[1 - I y]] + Pi)/(2 Pi)]) - 2 I Pi (Floor[(-Arg[(x + Sqrt[x^2 + 1])/(1 - I y)^(I/2)] - Arg[(I y + 1)^(I/2)] + Pi)/(2 Pi)] + Floor[(Pi - Im[Log[(x + Sqrt[x^2 + 1])/(1 - I y)^(I/2)]])/(2 Pi)] + Floor[(Pi - (1/2) Re[Log[I y + 1]])/(2 Pi)]) + Log[((x + Sqrt[x^2 + 1]) (I y + 1)^(I/2))/(1 - I y)^(I/2)]










Standard Form





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MathML Form







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</math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSinh", "[", "x_", "]"]], "-", RowBox[List["ArcTan", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["x", "+", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["x", "+", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["(", RowBox[List["x", "+", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]]]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", "1"]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List[RowBox[List["(", RowBox[List["x", "+", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List["Log", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", "1"]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["(", RowBox[List["x", "+", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", "1"]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]]]], "]"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02